How to define $A\uparrow B$ with a universal property as well as $A\oplus B$, $A\times B$, $A^B$ in category theory?

Let $a,b$ be objects of a closed symmetric monoidal category. Then $a^b$ may be written for the internal hom $\underline{\hom}(b,a)$. In fact, then we have the usual laws such as $a^{b+c}=a^b \times a^c$ and $(a^b)^c = a^{b \times c}$.

Now let us iterate this. $a^a = \underline{\hom}(a,a)$, $a^{a^a} = \underline{\hom}(\underline{\hom}(a,a),a)$, etc. We can define $^{n} a=a^{a^{a^{a^\dotsc}}}$ for every $n < \omega$. Assume that every object is dualizable (for example, consider the category of finite-dimensional vector spaces over some field), so that $a^b = b^* \otimes a$. Then one shows by induction that $$^n a = \left\{\begin{array}{c}(a^*)^{\otimes \frac{n}{2}} \otimes a^{\otimes \frac{n}{2}} & n \text{ even} \\ (a^*)^{\otimes \frac{n-1}{2}} \otimes a^{\otimes \frac{n+1}{2}} & n \text{ odd}\end{array}\right.$$ This case distinction indicates that it is impossible to give a natural definition of $^b a$ for objects $a,b$.